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1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
4 | <title>Laguerre (and Associated) Polynomials</title> | |
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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.sf_poly.laguerre"></a><a class="link" href="laguerre.html" title="Laguerre (and Associated) Polynomials">Laguerre (and Associated) | |
28 | Polynomials</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.sf_poly.laguerre.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.synopsis"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.synopsis">Synopsis</a> | |
33 | </h5> | |
34 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">laguerre</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
35 | </pre> | |
36 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
37 | ||
38 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
39 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
40 | ||
41 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
42 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
43 | ||
44 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
45 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
46 | ||
47 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
48 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
49 | ||
50 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
51 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> | |
52 | ||
53 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
54 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> | |
55 | ||
56 | ||
57 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
58 | </pre> | |
59 | <h5> | |
60 | <a name="math_toolkit.sf_poly.laguerre.h1"></a> | |
61 | <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.description"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.description">Description</a> | |
62 | </h5> | |
63 | <p> | |
64 | The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result | |
65 | type calculation rules</em></span></a>: note than when there is a single | |
66 | template argument the result is the same type as that argument or <code class="computeroutput"><span class="keyword">double</span></code> if the template argument is an integer | |
67 | type. | |
68 | </p> | |
69 | <p> | |
70 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
71 | be used to control the behaviour of the function: how it handles errors, | |
72 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
73 | documentation for more details</a>. | |
74 | </p> | |
75 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
76 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
77 | ||
78 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
79 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
80 | </pre> | |
81 | <p> | |
82 | Returns the value of the Laguerre Polynomial of order <span class="emphasis"><em>n</em></span> | |
83 | at point <span class="emphasis"><em>x</em></span>: | |
84 | </p> | |
85 | <p> | |
86 | <span class="inlinemediaobject"><img src="../../../equations/laguerre_0.svg"></span> | |
87 | </p> | |
88 | <p> | |
89 | The following graph illustrates the behaviour of the first few Laguerre Polynomials: | |
90 | </p> | |
91 | <p> | |
92 | <span class="inlinemediaobject"><img src="../../../graphs/laguerre.svg" align="middle"></span> | |
93 | </p> | |
94 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
95 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
96 | ||
97 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
98 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
99 | </pre> | |
100 | <p> | |
101 | Returns the Associated Laguerre polynomial of degree <span class="emphasis"><em>n</em></span> | |
102 | and order <span class="emphasis"><em>m</em></span> at point <span class="emphasis"><em>x</em></span>: | |
103 | </p> | |
104 | <p> | |
105 | <span class="inlinemediaobject"><img src="../../../equations/laguerre_1.svg"></span> | |
106 | </p> | |
107 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
108 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> | |
109 | </pre> | |
110 | <p> | |
111 | Implements the three term recurrence relation for the Laguerre polynomials, | |
112 | this function can be used to create a sequence of values evaluated at the | |
113 | same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>. | |
114 | </p> | |
115 | <p> | |
116 | <span class="inlinemediaobject"><img src="../../../equations/laguerre_2.svg"></span> | |
117 | </p> | |
118 | <p> | |
119 | For example we could produce a vector of the first 10 polynomial values using: | |
120 | </p> | |
121 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span> | |
122 | <span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> | |
123 | <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)).</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> | |
124 | <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> | |
125 | <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> | |
126 | </pre> | |
127 | <p> | |
128 | Formally the arguments are: | |
129 | </p> | |
130 | <div class="variablelist"> | |
131 | <p class="title"><b></b></p> | |
132 | <dl class="variablelist"> | |
133 | <dt><span class="term">n</span></dt> | |
134 | <dd><p> | |
135 | The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated. | |
136 | </p></dd> | |
137 | <dt><span class="term">x</span></dt> | |
138 | <dd><p> | |
139 | The abscissa value | |
140 | </p></dd> | |
141 | <dt><span class="term">Ln</span></dt> | |
142 | <dd><p> | |
143 | The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>. | |
144 | </p></dd> | |
145 | <dt><span class="term">Lnm1</span></dt> | |
146 | <dd><p> | |
147 | The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>. | |
148 | </p></dd> | |
149 | </dl> | |
150 | </div> | |
151 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
152 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> | |
153 | </pre> | |
154 | <p> | |
155 | Implements the three term recurrence relation for the Associated Laguerre | |
156 | polynomials, this function can be used to create a sequence of values evaluated | |
157 | at the same <span class="emphasis"><em>x</em></span>, and for rising degree <span class="emphasis"><em>n</em></span>. | |
158 | </p> | |
159 | <p> | |
160 | <span class="inlinemediaobject"><img src="../../../equations/laguerre_3.svg"></span> | |
161 | </p> | |
162 | <p> | |
163 | For example we could produce a vector of the first 10 polynomial values using: | |
164 | </p> | |
165 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span> | |
166 | <span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="comment">// order</span> | |
167 | <span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> | |
168 | <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)).</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> | |
169 | <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> | |
170 | <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> | |
171 | </pre> | |
172 | <p> | |
173 | Formally the arguments are: | |
174 | </p> | |
175 | <div class="variablelist"> | |
176 | <p class="title"><b></b></p> | |
177 | <dl class="variablelist"> | |
178 | <dt><span class="term">n</span></dt> | |
179 | <dd><p> | |
180 | The degree of the last polynomial calculated. | |
181 | </p></dd> | |
182 | <dt><span class="term">m</span></dt> | |
183 | <dd><p> | |
184 | The order of the Associated Polynomial. | |
185 | </p></dd> | |
186 | <dt><span class="term">x</span></dt> | |
187 | <dd><p> | |
188 | The abscissa value. | |
189 | </p></dd> | |
190 | <dt><span class="term">Ln</span></dt> | |
191 | <dd><p> | |
192 | The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>. | |
193 | </p></dd> | |
194 | <dt><span class="term">Lnm1</span></dt> | |
195 | <dd><p> | |
196 | The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>. | |
197 | </p></dd> | |
198 | </dl> | |
199 | </div> | |
200 | <h5> | |
201 | <a name="math_toolkit.sf_poly.laguerre.h2"></a> | |
202 | <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.accuracy"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.accuracy">Accuracy</a> | |
203 | </h5> | |
204 | <p> | |
205 | The following table shows peak errors (in units of epsilon) for various domains | |
206 | of input arguments. Note that only results for the widest floating point | |
207 | type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively | |
208 | zero error</a>. | |
209 | </p> | |
210 | <div class="table"> | |
211 | <a name="math_toolkit.sf_poly.laguerre.table_laguerre_n_x_"></a><p class="title"><b>Table 6.35. Error rates for laguerre(n, x)</b></p> | |
212 | <div class="table-contents"><table class="table" summary="Error rates for laguerre(n, x)"> | |
213 | <colgroup> | |
214 | <col> | |
215 | <col> | |
216 | <col> | |
217 | <col> | |
218 | <col> | |
219 | </colgroup> | |
220 | <thead><tr> | |
221 | <th> | |
222 | </th> | |
223 | <th> | |
224 | <p> | |
225 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
226 | </p> | |
227 | </th> | |
228 | <th> | |
229 | <p> | |
230 | GNU C++ version 5.1.0<br> linux<br> double | |
231 | </p> | |
232 | </th> | |
233 | <th> | |
234 | <p> | |
235 | GNU C++ version 5.1.0<br> linux<br> long double | |
236 | </p> | |
237 | </th> | |
238 | <th> | |
239 | <p> | |
240 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
241 | </p> | |
242 | </th> | |
243 | </tr></thead> | |
244 | <tbody><tr> | |
245 | <td> | |
246 | <p> | |
247 | Laguerre Polynomials | |
248 | </p> | |
249 | </td> | |
250 | <td> | |
251 | <p> | |
252 | <span class="blue">Max = 3.1e+003ε (Mean = 185ε)</span> | |
253 | </p> | |
254 | </td> | |
255 | <td> | |
256 | <p> | |
257 | <span class="blue">Max = 6.82ε (Mean = 0.408ε)</span><br> <br> | |
258 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.1e+03ε (Mean = 185ε)) | |
259 | </p> | |
260 | </td> | |
261 | <td> | |
262 | <p> | |
263 | <span class="blue">Max = 1.39e+04ε (Mean = 828ε)</span><br> | |
264 | <br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 4.2e+03ε (Mean | |
265 | = 251ε)) | |
266 | </p> | |
267 | </td> | |
268 | <td> | |
269 | <p> | |
270 | <span class="blue">Max = 1.39e+04ε (Mean = 828ε)</span> | |
271 | </p> | |
272 | </td> | |
273 | </tr></tbody> | |
274 | </table></div> | |
275 | </div> | |
276 | <br class="table-break"><div class="table"> | |
277 | <a name="math_toolkit.sf_poly.laguerre.table_laguerre_n_m_x_"></a><p class="title"><b>Table 6.36. Error rates for laguerre(n, m, x)</b></p> | |
278 | <div class="table-contents"><table class="table" summary="Error rates for laguerre(n, m, x)"> | |
279 | <colgroup> | |
280 | <col> | |
281 | <col> | |
282 | <col> | |
283 | <col> | |
284 | <col> | |
285 | </colgroup> | |
286 | <thead><tr> | |
287 | <th> | |
288 | </th> | |
289 | <th> | |
290 | <p> | |
291 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
292 | </p> | |
293 | </th> | |
294 | <th> | |
295 | <p> | |
296 | GNU C++ version 5.1.0<br> linux<br> double | |
297 | </p> | |
298 | </th> | |
299 | <th> | |
300 | <p> | |
301 | GNU C++ version 5.1.0<br> linux<br> long double | |
302 | </p> | |
303 | </th> | |
304 | <th> | |
305 | <p> | |
306 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
307 | </p> | |
308 | </th> | |
309 | </tr></thead> | |
310 | <tbody><tr> | |
311 | <td> | |
312 | <p> | |
313 | Associated Laguerre Polynomials | |
314 | </p> | |
315 | </td> | |
316 | <td> | |
317 | <p> | |
318 | <span class="blue">Max = 434ε (Mean = 11.1ε)</span> | |
319 | </p> | |
320 | </td> | |
321 | <td> | |
322 | <p> | |
323 | <span class="blue">Max = 0.84ε (Mean = 0.0358ε)</span><br> <br> | |
324 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 434ε (Mean = 10.7ε)) | |
325 | </p> | |
326 | </td> | |
327 | <td> | |
328 | <p> | |
329 | <span class="blue">Max = 167ε (Mean = 6.38ε)</span><br> <br> | |
330 | (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 206ε (Mean = 6.86ε)) | |
331 | </p> | |
332 | </td> | |
333 | <td> | |
334 | <p> | |
335 | <span class="blue">Max = 167ε (Mean = 6.38ε)</span> | |
336 | </p> | |
337 | </td> | |
338 | </tr></tbody> | |
339 | </table></div> | |
340 | </div> | |
341 | <br class="table-break"><p> | |
342 | Note that the worst errors occur when the degree increases, values greater | |
343 | than ~120 are very unlikely to produce sensible results, especially in the | |
344 | associated polynomial case when the order is also large. Further the relative | |
345 | errors are likely to grow arbitrarily large when the function is very close | |
346 | to a root. | |
347 | </p> | |
348 | <h5> | |
349 | <a name="math_toolkit.sf_poly.laguerre.h3"></a> | |
350 | <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.testing"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.testing">Testing</a> | |
351 | </h5> | |
352 | <p> | |
353 | A mixture of spot tests of values calculated using functions.wolfram.com, | |
354 | and randomly generated test data are used: the test data was computed using | |
355 | <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit | |
356 | precision. | |
357 | </p> | |
358 | <h5> | |
359 | <a name="math_toolkit.sf_poly.laguerre.h4"></a> | |
360 | <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.implementation"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.implementation">Implementation</a> | |
361 | </h5> | |
362 | <p> | |
363 | These functions are implemented using the stable three term recurrence relations. | |
364 | These relations guarantee low absolute error but cannot guarantee low relative | |
365 | error near one of the roots of the polynomials. | |
366 | </p> | |
367 | </div> | |
368 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
369 | <td align="left"></td> | |
370 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
371 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
372 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
373 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
374 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
375 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
376 | </p> | |
377 | </div></td> | |
378 | </tr></table> | |
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